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Fundamentals of Seismic Refraction
Theory, Acquisition, and Interpretation
Craig LippusManager, Seismic Products
Geometrics, Inc.
December 3, 2007
Geometrics, Inc.
• Owned by Oyo Corporation, Japan• In business since 1969• Seismographs, magnetometers, EM systems• Land, airborne, and marine• 80 employees
Located in San Jose, California
Fundamentals of Seismic Waves
Q. What is a seismic wave?
Fundamentals of Seismic Waves
A. Transfer of energy by way ofparticle motion.
Different types of seismic waves are characterized by their particle motion.
Q. What is a seismic wave?
Three different types of seismic waves
• Compressional (“p”) wave• Shear (“s”) wave• Surface (Love and Raleigh)
wave
Only p and s waves (collectively referred toas “body waves”) are of interest in seismic refraction.
Compressional (“p”) Wave
Identical to sound wave – particlemotion is parallel to propagationdirection.
Animation courtesy Larry Braile, Purdue University
Shear (“s”) Wave
Particle motion is perpendicularto propagation direction.
Animation courtesy Larry Braile, Purdue University
Velocity of Seismic Waves
Depends on density elastic moduli
3
4
KVp
Vs
where K = bulk modulus, = shear modulus, and = density.
Velocity of Seismic Waves
Bulk modulus = resistance to compression = incompressibility
Shear modulus = resistance to shear = rigidity
The less compressible a material is, the greater its p-wave velocity, i.e., sound travels about four times faster in water than in air. The more resistant a material is to shear, the greater its shear wave velocity.
Q. What is the rigidity of water?
A. Water has no rigidity. Its shear strength is zero.
Q. What is the rigidity of water?
Q. How well does water carry shear waves?
A. It doesn’t.
Q. How well does water carry shear waves?
Fluids do not carry shear waves. This knowledge, combined with earthquake observations, is what lead to the discovery that the earth’s outer core is a liquid rather than a solid – “shear wave shadow”.
p-wave velocity vs. s-wave velocity
p-wave velocity must always be greater than s-wave velocity. Why?
3
43
4
2
2
K
K
Vs
Vp
K and are always positive numbers, so Vp is always greater than Vs.
Velocity – density paradox
Q. We know that in practice, velocity tends to be directly proportional to density. Yet density is in the denominator. How is that possible?
Velocity – density paradox
A. Elastic moduli tend to increase with density also, and at a faster rate.
Q. We know that in practice, velocity tends to be directly proportional to density. Yet density is in the denominator. How is that possible?
Velocity – density paradox Note: Elastic moduli are important parameters for understanding rock properties and how they will behave under various conditions. They help engineers assess suitability for founding dams, bridges, and other critical structures such as hospitals and schools. Measuring p- and s-wave velocities can help determine these properties indirectly and non-destructively.
Q. How do we use seismic waves to understand the subsurface?
Q. How do we use seismic waves to understand the subsurface?
A. Must first understand wavebehavior in layered media.
Q. What happens when a seismic wave encounters a velocity discontinuity?
Q. What happens when a seismic wave encounters a velocity discontinuity?
A. Some of the energy is reflected, some is refracted.
We are only interested in refracted energy!!
Q. What happens when a seismic wave encounters a velocity discontinuity?
Five important concepts
• Seismic Wavefront• Ray• Huygen’s Principle• Snell’s Law• Reciprocity
Q. What is a seismic wavefront?
Q. What is a seismic wavefront?
A. Surface of constant phase, like ripples on a pond, but in three dimensions.
Q. What is a seismic wavefront?
The speed at which a wavefront travels is the seismic velocity of the material, and depends on the material’s elastic properties. In a homogenious medium, a wavefront is spherical, and its shape is distorted by changes in the seismic velocity.
Seismic wavefront
Q. What is a ray?
Q. What is a ray?
A. Also referred to as a “wavefrontnormal” a ray is an arrowperpendicular to the wave front,indicating the direction of travel atthat point on the wavefront. Thereare an infinite number of rays on awave front.
Ray
Huygens' Principle Every point on a wave front can be thought of as a new point source for waves generated in the direction the wave is traveling or being propagated.
Q. What causes refraction?
Q. What causes refraction?A. Different portions of the wave front reach the velocity boundary earlier than other portions, speeding up or slowing down on contact, causing distortion of wave front.
Understanding and Quantifying How Waves
Refract is Essential
Snell’s Law
2
1
sin
sin
V
V
r
i (1)
Snell’s Law
If V2>V1, then as i increases, r increases faster
Snell’s Lawr approaches 90o as i increases
Snell’s LawCritical Refraction
At Critical Angle of incidence ic, angle of refraction r = 90o
2
1
90sin
)sin(
V
Vic
2
1)sin(
V
Vic
2
11sinV
Vic
(2)
(3)
Snell’s LawCritical Refraction
At Critical Angle of incidence ic, angle of refraction r = 90o
Snell’s LawCritical Refraction
At Critical Angle of incidence ic, angle of refraction r = 90o
Snell’s LawCritical Refraction
Seismic refraction makes use of critically refracted, first-arrival energy only. The rest of the wave form is ignored.
Principal of Reciprocity
The travel time of seismic energy between two points is independent of the direction traveled, i.e., interchanging the source and the geophone will not affect the seismic travel time between the two.
Using Seismic Refraction to Map the Subsurface
Critical Refraction Plays a Key Role
11 /VxT
1212
V
df
V
cd
V
acT
)cos( ci
hdfac
)tan( cihdebc
)tan(2 cihxdebcxcd
2)(12
)tan(2
cos
2
V
ihx
iV
hT
c
c
22)(12
)tan(2
cos
2
V
x
V
ih
iV
hT
c
c
22)(12
)cos(
)sin(
cos
12
V
x
iV
i
iVhT
c
c
c
221
1
)(21
22
)cos(
)sin(
cos2
V
x
iVV
iV
iVV
VhT
c
c
c
221
122
)cos(
)sin(2
V
x
iVV
iVVhT
c
c
2
1sin
V
Vic (Snell’s Law)
221
1
2
12)cos(
)sin(2
V
x
iVV
iV
V
hVTc
c
22112
)cos(
)sin()sin(
1
2V
x
iVV
ii
hVTc
cc
212
)cos(2
V
x
V
ihT
c
221
2
12)cos()sin(
)(sin12
V
x
iiVV
ihVT
cc
c
221
2
12)cos()sin(
)(cos2
V
x
iiVV
ihVT
cc
c
222
)sin(
)cos(2
V
x
iV
ihT
c
c
)sin(21 ciVV
From Snell’s Law,
(4)
Using Seismic Refraction to Map the Subsurface
Using Seismic Refraction to Map the Subsurface
Using Seismic Refraction to Map the Subsurface
Using Seismic Refraction to Map the Subsurface
Using Seismic Refraction to Map the Subsurface
Using Seismic Refraction to Map the Subsurface
Using Seismic Refraction to Map the Subsurface
Using Seismic Refraction to Map the Subsurface
Using Seismic Refraction to Map the Subsurface
Using Seismic Refraction to Map the Subsurface
Using Seismic Refraction to Map the Subsurface
Using Seismic Refraction to Map the Subsurface
Using Seismic Refraction to Map the Subsurface
Using Seismic Refraction to Map the Subsurface
Using Seismic Refraction to Map the Subsurface
Using Seismic Refraction to Map the Subsurface
Using Seismic Refraction to Map the Subsurface
Using Seismic Refraction to Map the Subsurface
Using Seismic Refraction to Map the Subsurface
Using Seismic Refraction to Map the Subsurface
Depth{
12
12
2 VV
VVXcDepth
(5)
Using Seismic Refraction to Map the Subsurface
Depth{
For layer parallel to surface
12
12
2 VV
VVXcDepth
)cos(sin22
11
1
V
VVTi
(6)
212
)cos(2
V
x
V
ihT
c
12
12
2 VV
VVXch
2
11
1
sincos2VV
VTh
i
Summary of Important Equations
For refractor parallel to surface
2
1
sin
sin
V
V
r
i
2
11sinV
Vic
(2)
(3)
(1)
(5)
(4)
(6)
Snell’s Law
2
1)sin(
V
Vic
)cos(sin22
11
121
VV
VTh
i
1
32
2
21
3123
2
)/1cos(sin2
)/1cos(sin
)/1cos(sin
hVV
VVV
VVTT
h
ii
2143
1
32
421
2
211
411
24
3)/cos(sin2
)/cos(sin2)/cos(sin)/cos(sin
hhVV
VV
VhVVVV
TT
h
ii
Crossover Distance vs. Depth
Depth/Xc vs. Velocity Contrast
Important Rule of Thumb
The Length of the Geophone Spread Should be 4-5 times the depth of interest.
Dipping Layer
Defined as Velocity Boundary that is not Parallel to Ground Surface
You should always do a minimum of one shot at either end the spread. A single shot at one end does not tell you anything about dip, and if the layer(s) is dipping, your depth and velocity calculated from a single shot will be wrong.
Dipping Layer
If layer is dipping (relative to ground surface), opposing travel time curves will be asymmetrical.
Updip shot – apparent velocity > true velocityDowndip shot – apparent velocity < true velocity
Dipping Layer
Dipping Layer
)sin(sin2
11
11
1udc mVmVi
)sin(1 cd imV
)sin(1 cimuV
dc mVi 11sin
uc mVi 11sin
)sin(sin2
11
11
1ud mVmV
Dipping Layer
From Snell’s Law,
)sin(
12
ci
VV
cos)cos(2
1
c
iu
ui
TV
D
cos)cos(2
1
c
id
di
TV
D
Dipping Layer
The true velocity V2 can also be calculated by multiplying the harmonic mean of the up-dip and down-dip velocities by the cosine of the dip.
cos2
22
222
DU
DU
VV
VVV
What if V2 < V1?
2
1
sin
sin
V
V
r
i
What if V2 < V1?
Snell’s Law
2
1
sin
sin
V
V
r
i
What if V2 < V1?
Snell’s Law
If V1>V2, then as i increases, r increases, but not as fast.
What if V2 < V1?
If V2<V1, the energy refracts toward the normal.
None of the refracted energy makes it back to the surface.
This is called a velocity inversion.
Seismic Refraction requires that velocities increase with depth.
A slower layer beneath a faster layer will not be detected by seismic refraction.
The presence of a velocity inversion can lead to errors in depth calculations.
Delay Time Method
• Allows Calculation of Depth Beneath Each Geophone
• Requires refracted arrival at each geophone from opposite directions
• Requires offset shots
• Data redundancy is important
Delay Time Methodx
V1
V2
Delay Time Methodx
V1
V2
)cos(
)tan()tan(
)cos( 12221 c
BcBcA
c
AAB
iV
h
V
ih
V
ih
V
AB
iV
hT
Delay Time Methodx
)cos(
)tan()tan(
)cos( 12221 c
PcPcA
c
AAP
iV
h
V
ih
V
ih
V
AP
iV
hT
)cos(
)tan()tan(
)cos( 12221 c
BcBcA
c
AAB
iV
h
V
ih
V
ih
V
AB
iV
hT
V1
V2
Delay Time Methodx
)cos(
)tan()tan(
)cos( 12221 c
PcPcB
c
BBP
iV
h
V
ih
V
ih
V
BP
iV
hT
)cos(
)tan()tan(
)cos( 12221 c
PcPcA
c
AAP
iV
h
V
ih
V
ih
V
AP
iV
hT
)cos(
)tan()tan(
)cos( 12221 c
BcBcA
c
AAB
iV
h
V
ih
V
ih
V
AB
iV
hT
V1
V2
Delay Time Methodx
t T T TA P B P A B0
Definition:
V1
V2
(7)
ABBPAP TTTt 0
)cos(
)tan()tan(
)cos( 122210
c
PcPcA
c
A
iV
h
V
ih
V
ih
V
AP
iV
ht
)cos(
)tan()tan(
)cos( 12221 c
PcPcB
c
B
iV
h
V
ih
V
ih
V
BP
iV
h
)cos(
)tan()tan(
)cos( 12221 c
BcBcA
c
A
iV
h
V
ih
V
ih
V
AB
iV
h
2120
)tan(2
)cos(
2
V
ih
iV
h
V
ABBPAPt
cP
c
p
But from figure above, BPAPAB . Substituting, we get
2120
)tan(2
)cos(
2
V
ih
iV
h
V
BPAPBPAPt
cP
c
p
or
210
)tan(2
)cos(
2
V
ih
iV
ht
cP
c
p
)cos(
)sin(
)cos(
12
210
c
c
cp
iV
i
iVht
)cos(
)sin(
)cos(2
21
1
21
20
c
c
cp
iVV
iV
iVV
Vht
)cos(
)sin(
)cos(2
2121
1
2
10c
c
cp
iVV
i
iVVVV
Vht
2
1sin
V
VicSubstituting from Snell’s Law,
)cos(
)sin(
)cos(sin
1
22121
10c
c
c
cp
iVV
i
iVViVht
)cos(
)sin(
)cos(sin
1
22121
10c
c
c
cp
iVV
i
iVViVht
Multiplying top and bottom by sin(ic)
)cos()sin(
)(sin
)cos()sin(
12
21
2
2110
cc
c
ccp
iiVV
i
iiVVVht
)cos()sin(
)(cos2
21
2
10cc
cp
iiVV
iVht
)sin(
)cos(2
20
c
cp
iV
iht
)sin(
)cos(2
20
c
cp
iV
iht
2
1sin
V
Vic
Substituting from Snell’s Law,
10
)cos(2
V
iht
cp (8)
We get
11
)cos(
2
)cos(2
2 Ppoint at Delay time
V
ih
V
ihtD
cpcpoTP (9)
Reduced Traveltimes
Definition:
T’AP = “Reduced Traveltime” at point P for a source at A
T’AP=TAP’
x
Reduced traveltimes are useful for determining V2. A plot of T’ vs. x will be roughly linear, mostly unaffected by changes in layer thickness, and the slope will be 1/V2.
Reduced Traveltimesx
From the above figure, T’AP is also equal to TAP minus the Delay Time. From equation 9, we then get
2'
oAPTAPAP
tTDTT P
Reduced Traveltimesx
Earlier, we defined to as
t T T TA P B P A B0 Substituting, we get
22'
ABBPAPAP
oAPAP
TTTT
tTT
(7)
(10)
Reduced Traveltimes
T
T T TA P
A B A P B P'
2 2
Finally, rearranging yields
The above equation allows a graphical determination of the T’ curve. TAB is called the reciprocal time.
(11)
Reduced Traveltimes
TT T T
A PA B A P B P
'
2 2The first term is represented by the dotted line below:
Reduced Traveltimes
TT T T
A PA B A P B P
'
2 2The numerator of the second term is just the difference in the traveltimes from points A to P and B to P.
Reduced Traveltimes
TT T T
A PA B A P B P
'
2 2Important: The second term only applies to refracted arrivals. It does not apply outside the zone of “overlap”, shown in yellow below.
Reduced Traveltimes
TT T T
A PA B A P B P
'
2 2The T’ (reduced traveltime) curve can now be determined graphically by adding (TAP-TBP)/2 (second term from equation 9) to the TAB/2 line (first term from equation 9). The slope of the T’ curve is 1/V2.
We can now calculate the delay time at point P. From Equation 10, we see that
1
)cos(
2 V
iht cpo
According to equation 8
2'
oAPAP
tTT
1
0 )cos(
2'
V
ihT
tTT
cpAPAPAP
So
Now, referring back to equation 4
212
)cos(2
V
x
V
ihT
c
(12)
(4)
(8)
(10)
It’s fair to say that
21
)cos(2
V
x
V
ihT
cpAP
Combining equations 12 and 13, we get
1211
)cos()cos(2)cos('
V
ih
V
x
V
ih
V
ihTT
cpcpcpAPAP
Or
21
)cos('
V
x
V
ihT
cpAP
(13)
(14)
1
)cos(
V
ihD
cpTp
Referring back to equation 9, we see that
Substituting into equation 14, we get
221
)cos('
V
xD
V
x
V
ihT pT
cpAP
Or
2'
V
xTD APTp
hD V
iP
T
c
P
1
co s( )
Solving equation 9 for hp, we get
(15)
(16)
(9)
We know that the incident angle i is critical when r is 90o. From Snell’s Law,
2
1
sin
sin
V
V
r
i
2
1
90sin
sin
V
Vic
2
1sin
V
Vic
2
11sinV
Vic
Substituting back into equation 16,
)cos(
1
c
Tp
i
VDh
p
2
11
1
sincosVV
VDh
pTp
(16)
(17)
we get
In summary, to determine the depth to the refractor h at any given point p:
1.Measure V1 directly from the traveltime plot.
2.Measure the difference in traveltime to point P from opposing shots (in zone of overlap only).
3.Measure the reciprocal time TAB.
4. Per equation 11,
TT T T
A PA B A P B P
'
2 2
divide the reciprocal time TAB by 2.
,
5. Per equation 11,
TT T T
A PA B A P B P
'
2 2add ½ the difference time at each point P to TAB/2 to get the reduced traveltime at P, T’AP.
,
6. Fit a line to the reduced traveltimes, compute V2 from slope.
2'
V
xTD APTp
7. Using equation 15,
Calculate the Delay Time DT at P1, P2, P3….PN
(15)
8. Using equation 17,
Calculate the Depth h at P1, P2,
P3….PN
2
11
1
sincosVV
VDh
pTp (16)
That’s all there is to it!
More Data is Better Than Less
More Data is Better Than Less
More Data is Better Than Less
More Data is Better Than Less
More Data is Better Than Less
More Data is Better Than Less
More Data is Better Than Less
More Data is Better Than Less
More Data is Better Than Less
More Data is Better Than Less
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